Memory-augmented Transformers can implement Linear First-Order Optimization Methods

We show that memory-augmented Transformers (Memformers) can implement linear first-order optimization methods such as conjugate gradient descent, momentum methods, and more generally, methods that linearly combine past gradients. Building on prior work that demonstrates how Transformers can simulate preconditioned gradient descent, we provide theoretical and empirical evidence that Memformers can learn more advanced optimization algorithms. Specifically, we analyze how memory registers in Memformers store suitable intermediate attention values allowing them to implement algorithms such as conjugate gradient. Our results show that Memformers can efficiently learn these methods by training on random linear regression tasks, even learning methods that outperform conjugate gradient. This work extends our knowledge about the algorithmic capabilities of Transformers, showing how they can learn complex optimization methods.

Riemannian Bilevel Optimization

We develop new algorithms for Riemannian bilevel optimization. We focus in particular on batch and stochastic gradient-based methods, with the explicit goal of avoiding second-order information such as Riemannian hyper-gradients. We propose and analyze $\mathrm{RF^2SA}$, a method that leverages first-order gradient information to navigate the complex geometry of Riemannian manifolds efficiently. Notably, $\mathrm{RF^2SA}$ is a single-loop algorithm, and thus easier to implement and use. Under various setups, including stochastic optimization, we provide explicit convergence rates for reaching $\epsilon$-stationary points. We also address the challenge of optimizing over Riemannian manifolds with constraints by adjusting the multiplier in the Lagrangian, ensuring convergence to the desired solution without requiring access to second-order derivatives.